Thermochemical submodel

The energy equation, with temperature varying as a function of radial, circumferential, and axial position and time, is the basis of the thermochemical submodel. The energy absorbed or released during the cure or crystallization of the matrix is included in the energy balance. The appropriate multidimensional energy equation is Equation 13.1, 

For thermosets, Hu is the total heat of reaction of the matrix. For thermoplastics, Hu is the theoretical ultimate heat of crystallinity. 

Once the temperature history is known, the viscosities can be calculated from expressions of the form (Eq. 13.4) 

In order to complete the problem, the initial and boundary conditions must be given. The temperature and degree of cine or crystallinity must initially (at time zero) be specified at every point inside the composite and the mandrel. For the latter only the temperature is required. As boundary conditions, the temperatures or heat fluxes at the composite outside diameter and mandrel inner diameter must be specified. 

Solutions to these equations yield the temperature distribution inside the mandrel and inside the composite as a function of time. Degree of cure or crystallinity and matrix viscosity in the composite as a function of time are also determined. This model is the building block for the other submodels. Viscosity calculations are input to the fiber motion submodel. Temperature and cure calculations are input to the stress submodel. Temperature data are also input to the void submodel. 

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Thermochemical submodel The thermochemical submodel provides temperature, viscosity, degree of cure (for thermosets), crystallinity (for thermoplastics), and the time required to complete the cure process. 

The surface tension is found from an empirical formula and is a function of temperature (determined in the thermochemical submodel). The surrounding pressure P is determined in the resin flow or compaction submodels. The pressure within the void is determined by the partial pressures of the water vapor and air within the void. The mass of water vapor within the void changes during processing and can be described by Fickian diffusion across the void-composite interface [29], Once the mass of vapor inside the void and the pressure at the location are known, the change in void size can readily be calculated from Equation 13.19. Changes in void size are halted when the resin has solidified. 

Carlone and Palazzo (2008) developed a computational modelling of microwave assisted pultrusion. This model is based on an electromagnetic submodel, meant to evaluate the electric field distribution and the heat generation rate due to the microwave source and on a thermochemical submodel, used to determine the temperature and degree of cure distributions. The performed simulations revealed the relevance of design of the microwave cavity, the curing die, and the importance of the dielectric properties of the materials in microwave pultrusion process. 

Often there are cases where the submodels are poorly known or misunderstood, such as for chemical rate equations, thermochemical data, or transport coefficients. A typical example is shown in Figure 1 which was provided by David Garvin at the U. S. National Bureau of Standards. The figure shows the rate constant at 300°K for the reaction HO + O3 - HO2 + Oj as a function of the year of the measurement. We note with amusement and chagrin that if we were modelling a kinetics scheme which incorporated this reaction before 1970, the rate would be uncertain by five orders of magnitude As shown most clearly by the pair of rate constant values which have an equal upper bound and lower bound, a sensitivity analysis using such poorly defined rate constants would be useless. Yet this case is not atypical of the uncertainty in rate constants for many major reactions in combustion processes. 

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